Advertisement Policy, Reliability Dependent Imperfect Production and Environmental Pollution Control Problem in Fuzzy-Rough Environment

Abstract

In this study, a problem for advertising policy to avoid the depreciation rate of sale, an imperfect production rate and environmental pollution control is formulated with reliability and rework over a finite time horizon. The advertisement and production rates are function of time which are taken as control variables. Also, recent studies indicate that industrial solid waste (ISW) in production period has been an important contributor to greenhouse gas (GHG) emissions. The company has agreed to invest a large amount of money to manage the ISW through technological up gradation, etc that reduce the GHG in environment. The unit production cost is a function of production rate and also dependent on raw material cost, development cost due to technological up gradation and reliability and wear-tear cost. The total profit which consists of the sales proceeds, marginal selling price, production cost, operating cost, pollution cost, inventory holding cost and advertisement cost is formulated as an optimal control problem. The selling price and all inventory costs are fuzzy rough in nature and using expectation of fuzzy rough technique, the problem is converted into equivalent crisp problem and solved using Pontryagin's Maximum Principle and Generalized Reduced Gradient Method (GRG). Finally numerical experiment, sensitivity analysis for marginal selling price, defective parameter, rework and depreciation rate of sale and graphical representation are provided to illustrate the model.